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Every Wijsman topology relative to a Polish space is Polish
Author:
C. Costantini
Journal:
Proc. Amer. Math. Soc. 123 (1995), 2569-2574
MSC:
Primary 54B20; Secondary 54D65, 54E50
MathSciNet review:
1273484
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Additional Information
Abstract: Generalizing a result of G. Beer and a result of E. Effros, we show that if (X, d) is a separable and completely metrizable metric space, then the hyperspace of X endowed with the Wijsman topology is separable and completely metrizable.
- [Ba1]
A. Barbati, Strutture boreliane sugli iperspazi, Tesi di Laurea, Universitá di Milano, 1993.
- [Ba2]
Alberto
Barbati, The hyperspace of an analytic metrizable space is
analytic, Proceedings of the Eleventh International Conference of
Topology (Trieste, 1993), 1993, pp. 15–21 (1994) (English, with
English and Italian summaries). MR 1346314
(96c:54047)
- [Be]
Gerald
Beer, A Polish topology for the closed
subsets of a Polish space, Proc. Amer. Math.
Soc. 113 (1991), no. 4, 1123–1133. MR 1065940
(92c:54009), http://dx.doi.org/10.1090/S0002-9939-1991-1065940-6
- [BLLN]
Gerald
Beer, Alojzy
Lechicki, Sandro
Levi, and Somashekhar
Naimpally, Distance functionals and suprema of hyperspace
topologies, Ann. Mat. Pura Appl. (4) 162 (1992),
367–381. MR 1199663
(94c:54016), http://dx.doi.org/10.1007/BF01760016
- [CLZ]
C.
Costantini, S.
Levi, and J.
Ziemińska, Metrics that generate the same hyperspace
convergence, Set-Valued Anal. 1 (1993), no. 2,
141–157. MR 1239401
(94h:54010), http://dx.doi.org/10.1007/BF01027689
- [Ef]
Edward
G. Effros, Convergence of closed subsets in a
topological space, Proc. Amer. Math. Soc.
16 (1965),
929–931. MR 0181983
(31 #6208), http://dx.doi.org/10.1090/S0002-9939-1965-0181983-3
- [En]
Ryszard
Engelking, General topology, 2nd ed., Sigma Series in Pure
Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from
the Polish by the author. MR 1039321
(91c:54001)
- [FLL]
Sebastiano
Francaviglia, Alojzy
Lechicki, and Sandro
Levi, Quasiuniformization of hyperspaces and convergence of nets of
semicontinuous multifunctions, J. Math. Anal. Appl.
112 (1985), no. 2, 347–370. MR 813603
(87e:54025), http://dx.doi.org/10.1016/0022-247X(85)90246-X
- [LL]
A.
Lechicki and S.
Levi, Wijsman convergence in the hyperspace of a metric space,
Boll. Un. Mat. Ital. B (7) 1 (1987), no. 2,
439–451 (English, with Italian summary). MR 896334
(88e:54007)
- [Ba1]
- A. Barbati, Strutture boreliane sugli iperspazi, Tesi di Laurea, Universitá di Milano, 1993.
- [Ba2]
- -, The hyperspace of an analytic metrizable space is analytic, Proc. 11th International Conference on Topology (Trieste, September 1993); Rend. Ist. Mat. Trieste 25 (1993). MR 1346314 (96c:54047)
- [Be]
- G. Beer, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991), 1123-1133. MR 1065940 (92c:54009)
- [BLLN]
- G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. (4) 162 (1992), 367-381. MR 1199663 (94c:54016)
- [CLZ]
- C. Costantini, S. Levi, and J. Zieminska, Metrics that generate the same hyperspace convergence, Set-Valued Analysis 1 (1993), 141-157. MR 1239401 (94h:54010)
- [Ef]
- E. Effros, Convergence of closed subsets in a topological space, Proc. Amer. Math. Soc. 16 (1965), 929-931. MR 0181983 (31:6208)
- [En]
- R. Engelking. General topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
- [FLL]
- S. Francaviglia, A. Lechicki, and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. MR 813603 (87e:54025)
- [LL]
- A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Boll. Un. Mat. Ital. (7) B.1 (1987), 439-451. MR 896334 (88e:54007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1273484-5
PII:
S 0002-9939(1995)1273484-5
Keywords:
Wijsman topology,
Polish space,
completely metrizable space,
-subset
Article copyright:
© Copyright 1995 American Mathematical Society
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